Group rings and hyperbolic geometry

TL;DR

This paper develops an algorithm in group algebra for groups acting on hyperbolic spaces, demonstrating that certain ideals are free based on the action's minimal displacement, with various algebraic, geometric, and topological implications.

Contribution

It introduces a novel algorithm linking group actions on hyperbolic spaces to freeness of ideals in the group algebra, revealing new connections between geometry and algebra.

Findings

Ideals generated by few elements are free, depending on minimal displacement.

Establishes algebraic consequences of hyperbolic group actions.

Derives geometric and topological implications from the algebraic results.

Abstract

For a group acting on a hyperbolic space, we set up an algorithm in the group algebra showing that ideals generated by few elements are free, where few is a function of the minimal displacement of the action, and derive algebraic, geometric, and topological consequences.